DR. J. CASEY ON CYCLIDES AND SPHERO-QDARTICS. 
C99 
Demonstration. If the polar spheres of the three cyclides with respect to the three pairs 
of inverse points («', f3', y', V), («", /3", y", &"), /S'", S'") he X', Y', 71, X", Y", Z", 
X'", Y"', Z"', then the polar spheres of the cyclides with respect to the pair of inverse 
points whose sphero-coordinates are ka! -{-la" -\-ma"’, + ky' -\-ly" -\-my"', 
B' + lV'+mh" will plainly be kX'+lXl'+mX.'", kY'+lY"+mY'", kZ'+lZ”+mZ"'. 
Hence, eliminating k, I, m, we get the required locus, 
X', 
VI! 
X", 
X, 
Y", 
Y'", 
Z', 
Z", 
r/ll! 
5 
a cyclide of the sixth degree having the circle at infinity as a triple line. 
Cor. If the pair of inverse points move along a circle , the locus of the intersection of 
their polar spheres with respect to two cyclides will he a cyclide of the fourth degree. 
304. From article 301, we can easily infer theorems in cyclide reciprocation from 
known theorems in quadric reciprocation. Thus, if two spheres be concentric, the reci- 
procal of one with respect to the other is a concentric sphere. Hence, if two Cartesian 
cyclides having a common sphere of inversion have a common triple focus , the reciprocal 
of one with respect to the other is a Cartesian cyclide having the same triple focus; or, 
since the theorem concerning the spheres may be enunciated thus, if tangent planes 
to a sphere intersect at given angles, the locus of their point of intersection is a concentric 
sphere, and the envelope of the plane through their points of contact is another con- 
centric sphere. Hence we infer that if three generating spheres of a Cartesian cyclide 
intersect at given angles , the locus of their points of intersection is a Cartesian cyclide 
having the same triple focus , and the envelope of the sphere through their six points of 
contact is another Cartesian cyclide having also the same triple focus. These theorems 
may evidently be inferred by the methods of substitution given in the last Chapter. 
305. If we reciprocate one sphere with respect to another not concentric, we get a 
quadric of revolution. Hence the reciprocal of a Cartesian cyclide with respect to 
another Cartesian cyclide having a different triple focus is cl symmetrical cyclide , that is, 
a cyclide having one of its spheres of inversion opened out into a plane, the corresponding 
focal quadric being one of revolution. 
306. If we reciprocate a surface of revolution with respect 'to a sphere, we get a 
general quadric. Hence, if ive reciprocate a cyclide having a plane of symmetry with 
respect to a Cartesian cyclide, we get a general cyclide. 
307. The principles explained in recent articles will obviously give some of the 
systems of substitutions explained in the last Chapter ; and, conversely, the results of this 
Chapter may be derived from the substitutions of the last. It is unnecessary to pursue 
the subject further, and I shall conclude the section with the two following theorems: — 
1°. The locus of the intersection of three rectangular tangent planes to a quadric is a 
sphere. Hence the locus of the pairs of points common to three generating spheres of a 
cyclide which are mutually orthogonal is a Cartesian cyclide. 
mdccclxxi. 5 c 
